=+23.11. 20.7 23.101 Consider a persistent, irreducible Markov chain, and for a fixed state j let N ,, be the number of passages through j up to time n. Show that N/n - 1/m with probability 1, where m = Eg_ kff") is the mean return time

(replace 1 /m by 0 if this mean is infinite). See Lemma 3 in Section 8.